Higher spectral sequences

TL;DR

This paper introduces a higher spectral sequence framework for filtered chain complexes, extending previous constructions to include more differentials and unify various classical spectral sequences under this new structure.

Contribution

The author develops a generalized higher spectral sequence for multi-filtered complexes, expanding the differential structure and unifying several classical spectral sequences.

Findings

Enables more differentials in spectral sequences.

Unifies classical spectral sequences under a higher spectral sequence framework.

Provides a universal coefficient theorem analog for spectral sequences.

Abstract

In this article we construct what we call a higher spectral sequence for any chain complex (or topological space) that is filtered in $n$ compatible ways. For this we extend the previous spectral system construction of the author, and we show that it admits considerably more differentials than what was previously known. As a result, this endows the successive Leray--Serre, Grothendieck, chromatic--Adams--Novikov, and Eilenberg--Moore spectral sequences of the author with the structure of a higher spectral sequence. Another application is a universal coefficient theorem analog for spectral sequences.